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Popular >

derivative of x^ay^{1-a}

Solution

\frac{d}{d x} (x^{a} y^{1 - a})

Solution

a x^{a - 1} y^{1 - a} + y^{- a} (1 - a) \frac{d y}{d x} x^{a}

Solution steps

\frac{d}{d x} (x^{a} y^{1 - a})

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{a}, g = y^{1 - a}

= \frac{d}{d x} (x^{a}) y^{1 - a} + \frac{d}{d x} (y^{1 - a}) x^{a}

\frac{d}{d x} (x^{a}) = a x^{a - 1}

\frac{d}{d x} (y^{1 - a}) = y^{- a} (1 - a) \frac{d y}{d x}

= a x^{a - 1} y^{1 - a} + y^{- a} (1 - a) \frac{d y}{d x} x^{a}

Popular Examples

limit as x approaches 0 of arctan(5x)

x \to 0 lim (arctan (5 x))

d/(dt)((ln(t))^2)

\frac{d}{d t} ((ln (t))^{2})

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(1, - 1), (6, - 1)

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\int_{6}^{8} \frac{92}{( x - 6 ) ^{3}} d x

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\frac{d}{d x} (sin^{4} (cos (8 x)))

Frequently Asked Questions (FAQ)

What is the derivative of x^ay^{1-a} ?
The derivative of x^ay^{1-a} is ax^{a-1}y^{1-a}+y^{-a}(1-a)(dy)/(dx)x^a
What is the first derivative of x^ay^{1-a} ?
The first derivative of x^ay^{1-a} is ax^{a-1}y^{1-a}+y^{-a}(1-a)(dy)/(dx)x^a